A temporal logic of epistemic and normative justifications, with an application to the Protagoras paradox
Meghdad Ghari
TL;DR
This paper develops and analyzes a temporal justification logic, $\textsf{JTO}$, that combines linear temporal logic with epistemic and deontic justification to reason about time, knowledge, and obligation. It presents two sound and complete semantics—F-interpreted systems (based on Fitting models) and interpreted-neighborhood systems—to capture justification and its duals (permission) in time, along with a rigorous axiomatization and constant-specification framework. The formal apparatus is then applied to a detailed treatment of the Protagoras paradox, showing how explicit normative reasons and time-bound obligations can resolve apparent contradictions without invoking incompatible obligations, while distinguishing conventional vs juridical obligations. The study thus advances methodological tools for integrating time, knowledge, and normative reasoning and demonstrates a precise, logically consistent approach to classic deontic puzzles in a temporal setting.
Abstract
We combine linear temporal logic (with both past and future modalities) with a deontic version of justification logic to provide a framework for reasoning about time and epistemic and normative reasons. In addition to temporal modalities, the resulting logic contains two kinds of justification assertions: epistemic justification assertions and deontic justification assertions. The former presents justification for the agent's knowledge and the latter gives reasons for why a proposition is obligatory. We present two kinds of semantics for the logic: one based on Fitting models and the other based on neighborhood models. The use of neighborhood semantics enables us to define the dual of deontic justification assertions properly, which corresponds to the notion of permission in deontic logic. We then establish the soundness and completeness of an axiom system of the logic with respect to these semantics. Further, we formalize the Protagoras versus Euathlus paradox in this logic and present a precise analysis of the paradox, and also briefly discuss Leibniz's solution.